Question

(a) Prove that \(\int_{0}^{\pi / 2} \sin ^{2} x d x=\int_{0}^{x / 2} \cos ^{2} x d x\) by making the change of variable \(x=\frac{1}{2} \pi-t\) in one of the integrals. (b) Use the same method to prove that the averages of \(\sin ^{2}(n \pi x / l)\) and \(\cos ^{2}(n \pi x / l)\) are the same over a period.

Short answer

By changing the variable to \( x = \frac{\pi}{2} - t \), \( \int_{0}^{\pi/2} \sin^{2} x \, dx = \int_{0}^{\pi/2} \cos^{2}(x) \, dx \). Similarly, \( I_{sin} = I_{cos} \) over one period.

Step-by-step solution

- Original Integral Setup

Consider the integral \(I = \int_{0}^{\pi / 2} \sin^{2} x \, dx\).

- Change of Variable

Use the substitution \( x = \frac{\pi}{2} - t \). Consequently, \( dx = -dt \).

- Adjusting Limits

When \( x = 0 \), \( t = \frac{\pi}{2} \). And when \( x = \frac{\pi}{2} \), \( t = 0 \). Thus the integral becomes \(I = \int_{\pi/2}^{0} \sin^{2}(\frac{\pi}{2} - t) \, (-dt) \).

- Simplify the Integrand

Factor out the negative sign to switch the integration limits: \(I = \int_{0}^{\pi/2} \sin^{2}(\frac{\pi}{2} - t) \, dt \). Using the identity \( \sin (\frac{\pi}{2} - t) = \cos t \, \), the integrand simplifies to \(I = \int_{0}^{\pi/2} \cos^{2} t \, dt \).

- Conclusion for Part (a)

Therefore, \( \int_{0}^{\pi / 2} \sin^{2} x \, dx = \int_{0}^{\pi / 2} \cos^{2} t \, dt \). This completes the proof.

- Average Calculation Setup

To prove that the averages of \(\sin^{2}(n\pi x / l)\) and \(\cos^{2}(n\pi x / l)\) are the same, consider the interval over one period from \(0\) to \(l\).

- Integrate over One Period

Compute the integral of \(\sin^{2}(n\pi x / l)\): \(I_{sin} = \int_{0}^{l} \sin^{2} (n\pi x / l) \, dx \).

- Change of Variable for Cosine

Use the substitution \( x = l - t \), then compute \(I_{cos} = \int_{l}^{0} \cos^{2}(n\pi(l - t) / l) \, (-dt) \).

- Simplify for Cosine

Since \( \cos^{2}(n\pi (l - t) / l) = \cos^{2}(n\pi t / l) \, \), transform the integral back to: \(I_{cos} = \int_{0}^{l} \cos^{2}(n\pi x / l) \, dx \).

- Equal Averages

Thus, the integrals of \( \sin^{2}(n \pi x / l) \) and \(\cos^{2}(n \pi x / l) \) over one period are equal, proving that their averages are the same.

Key concepts

Change of Variables

In integration, a change of variables (also known as substitution) is a technique where we replace a variable in the integrand with a new variable. This method is useful to simplify the integral or to match the integrand to a known form.
To implement this method:

• Choose a new variable to substitute the original variable.
• Express the old variable and its differential in terms of the new variable.
• Adjust the integration limits if dealing with a definite integral.

For example, in the given exercise, we used the substitution \( x = \frac{\pi}{2} - t \). This changed the original integrand involving \( \sin^2 x \) to involve \( \cos^2 t \), which is easier to handle using known trigonometric identities.

Integration Techniques

There are several techniques to solve integrals, each suitable for different types of functions. Some common methods include:

• Substitution: Simplifies the integrand by changing the variable.
• Integration by parts: Breaks down complicated products of functions into simpler integrals.
• Partial fractions: Decomposes rational functions into simpler fractions to integrate.

In the example provided, we utilized the substitution method to change variables and simplify the integral. This technique is often the first step in tackling complex integrals.
When you choose the correct substitution, integration becomes straightforward, as we saw when transforming \( \sin^2 x \) into \( \cos^2 t \).

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the variable. They are essential tools in solving integrals involving trigonometric functions.
Some common trigonometric identities include:

• Pythagorean identities: \( \sin^2 x + \cos^2 x = 1 \)
• Angle sum and difference identities: \( \sin(a+b) = \sin a \cos b + \cos a \sin b \)
• Co-function identities: \( \sin(\frac{\pi}{2} - x) = \cos x \)

In part (a) of the exercise, we used the co-function identity \( \sin(\frac{\pi}{2} - t) = \cos t \) to transform the integral involving \( \sin^2 x \) into an integral involving \( \cos^2 t \). This simplified integration significantly.

Average Value of Functions

The average value of a function over an interval gives us an idea of the function's overall behavior within that interval. For a function \( f(x) \) over the interval \([a, b]\), the average value is given by:
\[ \text{Average value} = \frac{1}{b - a} \int_{a}^{b} f(x) \, dx \]
In part (b) of the exercise, we proved that the averages of \( \sin^2(n \pi x / l) \) and \( \cos^2(n \pi x / l) \) over one period are the same. By integrating these functions over one period \( [0, l] \) and showing that their integrals are equal, we demonstrated that their average values must also be equal.
In the context of the provided exercise:

• We computed the integral of \( \sin^2 (n\pi x / l) \).
• We then used a substitution to show the integral of \( \cos^2 (n\pi x / l) \) is the same.
• Since their integrals over one period are the same, their averages must be identical.

This demonstrates a key property of sine and cosine squared functions over a full period.